Work Time Problem

[coolhabhi]

The ratio of the number of days taken to complete a work by A,B,C is 6:3:2. Working together they complete a work in 20 days. If B left the work before 2 days and C left before 4 days to completion, in how many would the work be completed then?

a) 21(1/3) days
b) 21(2/3) days
c) 22(1/3) days
d) 22(2/3) days
e) 23 days

OE: D

[DavidG@VeritasPrep]

The ratio of the number of days taken to complete a work by A,B,C is 6:3:2. Working together they complete a work in 20 days. If B left the work before 2 days and C left before 4 days to completion, in how many would the work be completed then?

a) 21(1/3) days
b) 21(2/3) days
c) 22(1/3) days
d) 22(2/3) days
e) 23 days

OE: D

Let's say that A's rate is "x."
Because B takes half as many days to complete the job, we know that B's rate should be double A's. So call B's rate '2x.'
Because C takes 1/3 as many days to complete the job, we know that C's rate should be triple A's. So call C's rate '3x.'

The combined rate is x + 2x + 3x = 6x. We know that together, they can complete a job in 20 days, so their combined rate is 1 job/20 days or 1/20. If 6x = 1/20, then x = 1/120.

A's rate = x = 1/120
B's rate = 2x = 1/60
C's rate = 3x = 1/40

Let's say A works for T days. B will work for (T - 2) days and C will work for (T - 4) days
Amount of work performed by each

A: (1/120) * T
B: (1/60) * (T-2)
C: (1/40) * (T-4)

(1/120)T + (1/60)(T-2) + (1/40)(T - 4) = 1 job

Multiply through by 120

T + 2(T-2) + 3(T - 4) = 120
T + 2T-4 + 3T -12 = 120
6T -16 = 120
6T = 136
T = 136/6 = 22 2/3; Answer is D

[GMATGuruNY]

The ratio of the number of days taken to complete a work by A,B,C is 6:3:2. Working together they complete a work in 20 days. If B left the work before 2 days and C left before 4 days to completion, in how many would the work be completed then?

a) 21(1/3) days
b) 21(2/3) days
c) 22(1/3) days
d) 22(2/3) days
e) 23 days

Let the time ratio be applied to a job of 6 widgets.
If A takes 6 days to complete this job, A's rate = w/t = 6/6 = 1 widget per day.
If B takes 3 days to complete this job, B's rate = w/t = 6/3 = 2 widgets per day.
If C takes 2 days to complete this job, C's rate = w/t = 6/2 = 3 widgets per day.

Combined rate for A, B and C = 1+2+3 = 6 widgets per day.
Thus, the actual job = the amount of work produced by A, B and C in 20 days = (combined rate for A, B and C)(20 days) = 6*20 = 120 widgets.

Since C leaves 4 days early and B leaves 2 days early, A and B work together without C for 2 days, while A works alone for the last 2 days.
Amount of work produced by A and B in 2 days = (combined rate for A and B)(2 days) = (1+2)(2) = 6 widgets.
Amount of work produced by A in the last 2 days = (A's rate)(2 days) = 1*2 = 2 widgets.

Remaining work = 120 - 6 - 2 = 112 widgets.
The remaining work is produced by A, B and C working together.
Time for A, B and C to produce 112 widgets = w/(combined rate for A, B and C) = 112/6 = 56/3 days = 18 2/3 days.
Total time = (time when A, B and C all work together) + (time when A and B work together without C) + (time when A works alone) = (18 2/3) + 2 + 2 = 22 2/3 days.

The correct answer is D.